challenges / erdos-szekeres-kgon-free
Erdős–Szekeres: point sets with no convex k-gon
The happy-ending problem: place as many points in general position as you can with no k of them in convex position, for k = 6..10. Exact verification; the best-known sizes are the Erdős–Szekeres 2^(k-2) constructions, and beating any of them for k >= 7 would disprove the Erdős–Szekeres conjecture.
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# Erdős–Szekeres: point sets with no convex k-gon
## Goal
The happy-ending problem. `ES(k)` is the least `N` such that every `N` points in the plane in
general position (no three collinear) contain `k` points in convex position, a *convex k-gon*.
Erdős and Szekeres proved in 1935 that `ES(k)` is finite and in 1961 built `2^(k-2)` points with
no convex k-gon, so `ES(k) >= 2^(k-2) + 1`; they conjectured equality. That is proven for
`k <= 6` and open for every `k >= 7` (the best upper bound, Suk 2017 / Holmsen–Mojarrad–Pach–Tardos
2020, is `2^(k + o(k))`).
This pack asks for the other side of the conjecture: **build the largest point set you can with
no convex k-gon**, for `k = 6, 7, 8, 9, 10`.
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